How many lattice paths from $(0, 1, 2)$ to $(5, 5, 5)$ that pass through $(3, 3, 3)$ but not $(1, 2, 3)?$
My answer
Firstly try to get how many possible shortest lattice paths from $(0, 1, 2)$ to $(3, 3, 3)$ which I got as $120$. Then I found the number of shortest lattice paths from $(0, 1, 2)$ to $(1,2,3)$ which was $6$ and I subtracted $120$ from $6$ which would give the number of lattice paths from $(0,1,2)$ to $(3,3,3)$ that does go through $(1,2,3)$ (equal to $114$). Lastly found the number of lattice paths from $(3,3,3)$ to $(5,5,5)$ which was $120$ and added $114$ which would give $234$ lattice paths.
In many places you have added while multiplication was required.
To find the number of lattice paths from $(0, 1, 2)$ to $(3, 3, 3)$, you have $6$ steps of which $3$ must be in one direction, $2$ must be in the other and the remaining $1$ along the $z$ axis. To choose the three along the $x$ axis, there are ${6\choose 3}=20$ ways. The $2$ along the $y$ axis can be chosen from the remaining $3$ in ${3\choose 2}=3$ ways. The remaining step must be along the $z$ axis. Thus, we get a total of $20\times 3=60$ ways.
Of these, we need to count the number of paths which pass through the point $(1, 2, 3)$. Firstly, count the number of paths from $(0, 1, 2)$ to $(1, 2, 3)$. This is equal to the number of orders in which one can go one step in each of the three directions, $3!=6$. The number of ways to get from $(1, 2, 3)$ to $(3, 3, 3)$ can be calculated as follows:
There are $3$ steps of which one is along the $y$ axis and the others are along the $x$ axis. The path along the $y$ axis can be chosen in $3$ ways. This gives $3$ ways to go from $(1, 2, 3)$ to $(3, 3, 3)$.
Thus, the total number of lattice paths from $(0, 1, 2)$ to $(3, 3, 3)$ passing through $(1, 2, 3)$ is equal to $6\times 3=18.$ Subtracting this from the total number of paths from $(0, 1, 2)$ to $(3, 3, 3)$, we get $60-18=42$ paths.
The number of paths from $(3, 3, 3)$ to $(5, 5, 5)$ is calculated as follows:
Of the path of length $6$, $2$ are along the $x$ axis. These can be chosen in ${6\choose 2}=15$ ways. Then of the remaining $4,$ $2$ must be chosen to be along the $y$ axis in ${4\choose 2}=6$ ways. The remaining part of the path is along the $y$ axis. Thus, there are a total of $15\times 6=90$ ways.
Multiplying, we get a total of $42\times 90=3780$ shortest paths from $(0, 1, 2)$ to $(5, 5, 5)$ which pass through $(3, 3, 3)$ but not through $(1, 2, 3)$.